It is proved in another post that the product and coproduct do not exist in the category of totally ordered sets (except in some trivial cases). (In this post I will only consider the category TOrd, so that the morphisms are the monotone maps, not the strictly monotone maps.)
It seems to me nevertheless that there is a natural way to define the "coproduct" and "product" of totally ordered sets $A$ and $B$ if you do not insist on it being symmetric. In other words, we will have $A \oplus B \not \cong B \oplus A$ and $A \times B \not \cong B \times A$.
For an asymmetric direct sum (coproduct), let $A \oplus B$ be the disjoint union of $A$ and $B$, where we define $a \le b$ for all $a \in A$, $b \in B$.
For an asymmetric direct product, let $A \times B$ be the cartesian product of ordered pairs, and let $(a, b) \le (a', b')$ whenever $a < a'$ OR $a = a'$ and $b \le b'$. (I.e., the order is lexicographic).
Can we come up with a universal mapping property which is satisfied by either of these constructions?
I understand that it might seem to be wishful thinking to expect these seeming poorly-behaved constructions to satisfy a category-theoretic property. So here is some motivation for why I think they should satisfy such a property.
A similar asymmetric construction works in other categories as well. In the category of well-ordered sets, the exact same construction is used for sum and product in ordinal arithmetic. In the category of monoids, the disjoint union of $A$ and $B$ can be a monoid as well if you say that $ab = b$ for any $a \in A$, $b \in B$. The construction agrees with the coproduct and product in the category of sets, although in this case of course both are commutative.
Both of these definitions would naturally generalize to taking a direct sum or product of any finite or infinite number of tosets.
We have associativity: $A \oplus (B \oplus C) \cong A \oplus (B \oplus C)$ and similarly for $\times$.
We also have $A \oplus 0 \cong 0 \oplus A \cong A$ and $1 \times A \cong A \times 1 \cong A$, where $0$ and $1$ are the initial and terminal objects in TOrd.
My understanding of category theory is mostly limited to its applications in algebra, in particular in groups, commutative rings, and modules.